Extension theory and the calculus of butterflies

This paper provides a unified treatment of two distinct viewpoints concerning the classification of group extensions: the first uses weak monoidal functors, the second classifies extensions by means of suitable H2 -actions. We develop our theory formally, by making explicit a connection between (non-abelian) G-torsors and fibrations. Then we apply our general framework to the classification of extensions in a semi-abelian context, by means of butterflies between internal crossed modules. As a main result, we get an internal version of Dedecker's theorem on the classification of extensions of a group by a crossed module. In the semi-abelian context, Bourn's intrinsic Schreier–Mac Lane extension theorem turns out to be an instance of our Theorem 6.3. Actually, even just in the case of groups, our approach reveals a result slightly more general than classical Schreier–Mac Lane theorem.